Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Error Analysis of Fully Discrete Scheme for the Cahn–Hilliard–Magneto-Hydrodynamics Problem

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper we analyze a fully discrete scheme for a general Cahn–Hilliard equation coupled with a nonsteady Magneto-hydrodynamics flow, which describes two immiscible, incompressible and electrically conducting fluids with different mobilities, fluid viscosities and magnetic diffusivities. A typical fully discrete scheme, which is comprised of conforming finite element method and the Euler semi-implicit discretization based on a convex splitting of the energy of the equation is considered in detail. We prove that our scheme is unconditionally energy stable and obtain some optimal error estimates for the concentration field, the chemical potential, the velocity field, the magnetic field and the pressure. The results of numerical tests are presented to validate the rates of convergence.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data Availability

Enquiries about data availability should be directed to the author.

References

  1. Anderson D.M., McFadden G.B., Wheeler A.A.: Diffuse-interface methods in fluid mechanics, Annual Review of Fluid Mechanics, vol. 30, pp. 139–165. Annual Reviews, Palo Alto, CA (1998)

  2. Bermudez, A., Munoz-Sola, R., Vazquez, R.: Analysis of two stationary magneto-hydrodynamic systems of equations including Joule heating. J. Math. Anal. Appl. 368, 444–468 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cai, Y., Shen, J.: Error estimates for a fully discretized scheme to a Cahn–Hilliard phase-field model for two-phase incompressible flows. Math. Comput. 313, 2057–2090 (2018)

    MathSciNet  MATH  Google Scholar 

  4. Davidson, P.: An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  5. Diegel, A., Feng, X., Wise, S.M.: Analysis of a mixed finite element method for a Cahn–Hilliard–Darcy–Stokes system. SIAM J. Numer. Anal. 53, 127–152 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Elliott, C.M., French, D.A., Milner, F.A.: A second order splitting method for the Cahn–Hilliard equation. Numer. Math. 54, 575–590 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Feng, X., Prohl, A.: Error analysis of a mixed finite element method for the Cahn–Hilliard equation. Numer. Math. 99, 47–84 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feng, X.: Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44, 1049–1072 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feng, X., He, Y., Liu, C.: Analysis of finite element approximations of a phase field model for two-phase fluids. Math. Comput. 258, 539–571 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Feng, X., Li, Y., Xing, Y.: Analysis of mixed interior penalty discontinuous Galerkin methods for the Cahn–Hilliard equation and the Hele–Shaw flow. SIAM J. Numer. Anal. 54, 825–847 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gurtin, M.E., Polignone, D., Vinals, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6, 815–831 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  12. Girault, V., Raviart, P.A.: Finite Element Methods for Navier–Stokes Equations. Series in Computational Mathematics. Springer, New York (1986)

    Book  MATH  Google Scholar 

  13. Gunzburger, M.D., Meir, A.J., Peterson, J.S.: On the existence and uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics. Math. Comput. 56, 523–563 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454, 2617–2654 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gerbeau, J.F., Le Bris, C., Lelièvre, T.: Mathematical Methods for the Magnetohydrodynamics of Liquid Metals. Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2006)

    MATH  Google Scholar 

  16. Logg, A., Mardal, K.-A., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  17. Han, D., Wang, X., Wu, H.: Existence and uniqueness of global weak solutions to a Cahn–Hilliard–Stokes–Darcy system for two phase incompressible flows in karstic geometry. J. Differ. Equ. 257, 3887–3933 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. He, Y.: Unconditional convergence of the Euler semi-implicit scheme for the three-dimensional incompressible MHD equations. IMA J. Numer. Anal. 35, 767–801 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kim, S., Lee, E.B., Choi, W.: Newton’s algorithm for magnetohydrodynamic equations with the initial guess from Stokes-like problem. J. Comput. Appl. Math. 309, 1–10 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lorca, S.A., Boldrini, J.L.: The initial value problem for a generalized Boussinesq model. Nonlinear Anal. 36, 457–480 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179, 211–228 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Layton, W., Tran, W., Trenchea, H.: Stability of partitioned methods for magnetohydrodynamics flows at small magnetic Reynolds number. Contemp. Math. 586, 231–238 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moreau, R.: Magneto-Hydrodynamics. Kluwer, Dordrecht (1990)

    Book  Google Scholar 

  24. Morley, N.B., Smolentsev, S., Barleon, L., Kirillov, I.R., Takahashi, M.: Liquid magnetohydrodynamics-recent progress and future directions for fusion. Fusion Eng. Des. 51–52, 701–713 (2000)

    Article  Google Scholar 

  25. Prohl, A.: Convergent finite element discretizations of the nonstationary incompressible magnetohydrodynamic system. Math. Model. Numer. Anal. 42, 1065–1087 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Szekely, J.: Fluid Flow Phenomena in Metals Processing. Academic Press, New York (1979)

    Google Scholar 

  27. Shen, J., Yang, X.: Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. SIAM J. Numer. Anal. 53, 279–296 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Temam R.: Navier-Stokes Equations: Theory and Numerical analysis, 3rd ed., Studies in Mathematics and Its Applications, vol. 2, North-Holland Publishing Co., Amsterdam (1984)

  29. Tabata, M., Tagami, D.: Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients. Numer. Math. 100, 351–372 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tone, F.: On the long-time \(H^2\)-stability of the implicit Euler scheme for the 2D magnetohydrodynamics equations. J. Sci. Comput. 38, 331–348 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Wiedmer, M.: Finite element approximation for equations of magnetohydrodynamics. Math. Comput. 69, 83–101 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yue, P., Feng, J.J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang, J., Mao, S., He, X., Yang, X., He, Y.: A diffuse interface model and semi-implicit energy stable finite element method for two-phase magnetohydrodynamic flows. Comput. Methods Appl. Mech. Eng. 356, 435–464 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work is supported by the Natural Science Foundation of China (No. 11701498).

Author information

Authors and Affiliations

  1. School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, 224051, China

    Hailong Qiu

Authors
  1. Hailong Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hailong Qiu.

Ethics declarations

Conflict of interest

The author declare no competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiu, H. Error Analysis of Fully Discrete Scheme for the Cahn–Hilliard–Magneto-Hydrodynamics Problem. J Sci Comput 95, 16 (2023). https://doi.org/10.1007/s10915-023-02147-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-023-02147-z

Keywords

Mathematics Subject Classification

Search

Navigation